\documentclass[a4paper]{article}
\usepackage[margin=1in]{geometry}
\usepackage{ctex}
\usepackage{tikz}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{xltxtra}
\usepackage{mflogo,texnames}
\usepackage{graphicx}
\usetikzlibrary{arrows.meta}

\title{\heiti\zihao{2} 习题1.5}
\author{中书君}
\date{\songti 2021年1月13日}

\begin{document}
\maketitle
\section{将下列各点的极坐标化为直角坐标:
  $$
      \left(\sqrt{2}, \frac{\pi}{4}\right) ;\left(6,-\frac{\pi}{3}\right);(5, \pi)
  $$}
\textbf{解}\quad
$$
    (1,1);(3,-\sqrt{3});(-5,0)
$$

\section{将下列各点的直角坐标化为极坐标:
  $$
      (-1,-1) ;(0,-5) ;(-\sqrt{3}, 1)
  $$}
\textbf{解}\quad
$$
    (\sqrt{2},-\frac{3}{4}\pi);(5,-\frac{\pi}{2});(2,\frac{5}{6}\pi)
$$

\section{求出极坐标方程 $r=\sin \theta+2 \cos \theta$ 所表示的曲线.}
\textbf{解}\quad
$x^{2}+y^{2}-(2x+y)=0$,$(x-1)^{2}+(y-\frac{1}{2})^{2}=\frac{5}{4}$.

\section{画出心形线 $r=a(1+\cos \theta)$ (其中 $a>0$) 的示意图.}
\begin{tikzpicture}[samples=200]
    \draw[-Stealth](-0.5,0)--(2.5,0)node[below]{$x$};
    \draw[-Stealth](0,-1.5)--(0,1.5)node[left]{$y$};
    \draw[domain=-pi:pi]plot({1.6*cos(\x/2 r)*cos(\x r)},{1.6*cos(\x/2 r)*sin(\x r)});
    \node at(-5pt,-5pt){$O$};\node[below]at(1.8,0){$2a$};
    \node at(1.5,-1.4){$r=a(1+\cos\theta)$};
\end{tikzpicture}

\end{document}